ANALYTIC FUNCTIONS OF SEVERAL COMPLEX

VARIABLES

ROBERT C. GUNNING HUGO ROSSI

AMS CHELSEA PUBLISHINGAmerican Mathematical Society • Providence, Rhode Island

ANALYTIC FUNCTIONS OF SEVERAL COMPLEX

VARIABLES

ROBERT C. GUNNING

HUGO ROSSI

AMS CHELSEA PUBLISHINGAmerican Mathematical Society • Providence, Rhode Island

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http://dx.doi.org/10.1090/chel/368.H

2010 Mathematics Subject Classification. Primary 32C15, 32C22, 32C35, 32Exx, 32Q28.

For additional information and updates on this book, visitwww.ams.org/bookpages/chel-368

Library of Congress Cataloging-in-Publication Data

Gunning, R. C. (Robert Clifford), 1931–Analytic functions of several complex variables / Robert C. Gunning, Hugo Rossi.

p. cm.Originally published: Englewood Cliffs, N.J. : Prentice-Hall, 1965, in series: Prentice-Hall

series in modern analysis.Includes bibliographical references and index.ISBN 978-0-8218-2165-7 (alk. paper)1. Functions of several complex variables. 2. Functional analysis. I. Rossi, Hugo. II. Title.

QA331.G78 2009515′.94—dc22

2009025418

Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Acquisitions Department, American Mathematical Society,201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made bye-mail to reprint-permission@ams.org.

c© 1965 held by the American Mathematical Society. All rights reserved.Reprinted by the American Mathematical Society, 2009, 2015.

The American Mathematical Society retains all rightsexcept those granted to the United States Government.

Printed in the United States of America.

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Preface to the AMS/Chelsea Edition

We were led to write this book after having given joint graduate courses onfunctions of several complex variables at Princeton in the early 1960’s. At thattime the subject was in a very active and transitional period, with a good dealof new material beyond what had been covered in the classical books then avail-able, especially Behnke and Thullen’s Theorie der Funktionen Mehrer KomplexerVeranderlichen of 1933 and Bochner and Martin’s Several Complex Variables of1948. Lars Hormander’s book An Introduction to Complex Analysis in Several Vari-ables appeared very shortly after our book, and focused on the approach throughpartial differential equations. In the subsequent years the subject has expandedvastly, with remarkable achievements in a number of directions – integral formulas,properties and applications of the ∂ operator, detailed properties of holomorphicvarieties both algebraic and topological, further detailed properties of sheaves andholomorphic mappings, and so on; and there have been a great many excellent booksin the intervening years, covering the whole range of new results and techniquesas well as providing some very useful introductions to the topic from a variety ofpoints of view. In view of all these further sources of information about the sub-ject, it is perhaps natural to ask why this book should be reissued at this point,particularly since we are not attempting to rework the book to correct a perhapssurprising number of errors and misprints. We may hope that a review of the stateof the subject a half-century ago might be of some historical interest, and in addi-tion that a short survey, focusing on the problems and techniques flourishing at thetime of its writing, might serve as another useful introduction to the subject and apreparation for embarking on more detailed reading of the extensive literature thathas arisen.

We wish to thank the American Mathematical Society, and particularly EdwardDunne, for their interest in reissuing the book and willingness to take the originalas it stands. They are providing a web site, to which we would like to encourageany readers there may be to submit corrections and modifications, for the use ofothers who might like to take up a beautiful and very active topic in mathematics.The URL for this website is:

http://www.ams.org/bookpages/chel-368/.

This website includes a bibliography of comprehensive texts on several complexvariables. Readers are encouraged to suggest entries that do not yet appear on thislist.

Robert C. GunningHugo Rossi

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CHEL/368.H